|
Aitken's Delta-squared Process
In numerical analysis, Aitken's delta-squared process or Aitken extrapolation is a series acceleration method used for accelerating the rate of convergence of a sequence. It is named after Alexander Aitken, who introduced this method in 1926 as part of an extension to Bernoulli's method. It is most useful for accelerating the convergence of a sequence that is converging linearly. A precursor form was known to Seki Kōwa (1642 – 1708) and applied to the rectification of the circle, i.e., to the calculation of π. Definition Given a sequence X = with n = 0, 1, 2, 3, \ldots, Aitken's delta-squared process associates to this sequence the new sequence A = (a_n) = , which can also be written as A = \left( x_n-\frac \right), with \Delta x_= x_-x_ and \Delta^2 x_n=x_n -2x_ + x_=\Delta x_-\Delta x_. Both are the same sequence algebraically but the latter has improved numerical stability in computational implementation. A /math> is ill-defined if the sequence \Delta^2 = (\Delt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt to find approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences like economics, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator (mathematics)
In mathematics, an operator is generally a Map (mathematics), mapping or function (mathematics), function that acts on elements of a space (mathematics), space to produce elements of another space (possibly and sometimes required to be the same space). There is no general definition of an ''operator'', but the term is often used in place of ''function'' when the domain of a function, domain is a set of functions or other structured objects. Also, the domain of an operator is often difficult to characterize explicitly (for example in the case of an integral operator), and may be extended so as to act on related objects (an operator that acts on functions may act also on differential equations whose solutions are functions that satisfy the equation). (see Operator (physics) for other examples) The most basic operators are linear maps, which act on vector spaces. Linear operators refer to linear maps whose domain and range are the same space, for example from \mathbb^n to \mathbb^n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shanks Transformation
In numerical analysis, the Shanks transformation is a non-linear series acceleration method to increase the rate of convergence of a sequence. This method is named after Daniel Shanks, who rediscovered this sequence transformation in 1955. It was first derived and published by R. Schmidt in 1941. Formulation For a sequence \left\_ the series :A = \sum_^\infty a_m\, is to be determined. First, the partial sum A_n is defined as: :A_n = \sum_^n a_m\, and forms a new sequence \left\_. Provided the series converges, A_n will also approach the limit A as n\to\infty. The Shanks transformation S(A_n) of the sequence A_n is the new sequence defined byBender & Orszag (1999), pp. 368–375.Van Dyke (1975), pp. 202–205. :S(A_n) = \frac = A_ - \frac where this sequence S(A_n) often converges more rapidly than the sequence A_n. Further speed-up may be obtained by repeated use of the Shanks transformation, by computing S^2(A_n)=S(S(A_n)), S^3(A_n)=S(S(S(A_n))), etc. Note that the non ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequence Transformation
In mathematics, a sequence transformation is an Operator (mathematics), operator acting on a given space of sequences (a sequence space). Sequence transformations include linear mappings such as convolution, discrete convolution with another sequence and resummation of a sequence and nonlinear mappings, more generally. They are commonly used for series acceleration, that is, for improving the rate of convergence of a slowly convergent sequence or series (mathematics), series. Sequence transformations are also commonly used to compute the antilimit of a divergent series numerically, and are used in conjunction with extrapolation methods. Classical examples for sequence transformations include the binomial transform, Möbius transform, and Stirling transform. Definitions For a given sequence :(s_n)_,\, and a sequence transformation \mathbf, the sequence resulting from transformation by \mathbf is :\mathbf( ( s_n ) ) = ( s'_n )_, where the elements of the transformed sequence a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richardson Extrapolation
In numerical analysis, Richardson extrapolation is a Series acceleration, sequence acceleration method used to improve the rate of convergence of a sequence of estimates of some value A^\ast = \lim_ A(h). In essence, given the value of A(h) for several values of h, we can estimate A^\ast by extrapolating the estimates to h=0. It is named after Lewis Fry Richardson, who introduced the technique in the early 20th century, though the idea was already known to Christiaan Huygens in Christiaan_Huygens#De_Circuli_Magnitudine_Inventa, his calculation of \pi. In the words of Garrett Birkhoff, Birkhoff and Gian-Carlo Rota, Rota, "its usefulness for practical computations can hardly be overestimated."Page 126 of Practical applications of Richardson extrapolation include Romberg integration, which applies Richardson extrapolation to the trapezoid rule, and the Bulirsch–Stoer algorithm for solving ordinary differential equations. General formula Notation Let A_0(h) be an approximation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fixed Point Iteration
In numerical analysis, fixed-point iteration is a method of computing fixed point (mathematics), fixed points of a function. More specifically, given a function f defined on the real numbers with real values and given a point x_0 in the domain of a function, domain of f, the fixed-point iteration is x_=f(x_n), \, n=0, 1, 2, \dots which gives rise to the sequence x_0, x_1, x_2, \dots of iterated function applications x_0, f(x_0), f(f(x_0)), \dots which is hoped to limit (mathematics), converge to a point x_\text. If f is continuous, then one can prove that the obtained x_\text is a fixed point of f, i.e., f(x_\text)=x_\text . More generally, the function f can be defined on any metric space with values in that same space. Examples * A first simple and useful example is the Babylonian method for computing the square root of , which consists in taking f(x) = \frac 1 2 \left(\frac a x + x\right), i.e. the mean value of and , to approach the limit x = \sqrt a (from whatever startin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit Of A Sequence
As the positive integer n becomes larger and larger, the value n\times \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n \times \sin\left(\tfrac1\right) equals 1." In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the \lim symbol (e.g., \lim_a_n).Courant (1961), p. 29. If such a limit exists and is finite, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests. Limits can be defined in any metric space, metric or topological space, but are usually first encountered in the real numbers. History The Greek philosopher Zeno of Elea is famous for formulating Zeno's paradoxes, paradoxes that involve limiting processes. Leucippus, Democritus, Antiphon (person), Antiphon, Eudoxus of Cnidus, Eudoxus, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rate Of Convergence
In mathematical analysis, particularly numerical analysis, the rate of convergence and order of convergence of a sequence that converges to a limit are any of several characterizations of how quickly that sequence approaches its limit. These are broadly divided into rates and orders of convergence that describe how quickly a sequence further approaches its limit once it is already close to it, called asymptotic rates and orders of convergence, and those that describe how quickly sequences approach their limits from starting points that are not necessarily close to their limits, called non-asymptotic rates and orders of convergence. Asymptotic behavior is particularly useful for deciding when to stop a sequence of numerical computations, for instance once a target precision has been reached with an iterative root-finding algorithm, but pre-asymptotic behavior is often crucial for determining whether to begin a sequence of computations at all, since it may be impossible or imprac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Methods Of Computing Square Roots
Square root algorithms compute the non-negative square root \sqrt of a positive real number S. Since all square roots of natural numbers, other than of perfect squares, are irrational, square roots can usually only be computed to some finite precision: these algorithms typically construct a series of increasingly accurate approximations. Most square root computation methods are iterative: after choosing a suitable initial estimate of \sqrt, an iterative refinement is performed until some termination criterion is met. One refinement scheme is Heron's method, a special case of Newton's method. If division is much more costly than multiplication, it may be preferable to compute the inverse square root instead. Other methods are available to compute the square root digit by digit, or using Taylor series. Rational approximations of square roots may be calculated using continued fraction expansions. The method employed depends on the needed accuracy, and the available tools and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steffensen's Method
In numerical analysis, Steffensen's method is an iterative method for numerical root-finding named after Johan Frederik Steffensen that is similar to the secant method and to Newton's method. Steffensen's method achieves a quadratic order of convergence without using derivatives, whereas the more familiar Newton's method also converges quadratically, but requires derivatives and the secant method does not require derivatives but also converges less quickly than quadratically. Steffensen's method has the drawback that it requires two function evaluations per step, whereas the secant method requires only one evaluation per step, so it is not necessarily most efficient in terms of computational cost, depending on the number of iterations each requires. Newton's method also requires evaluating two functions per step – for the function and for its derivative – and its computational cost varies between being the same as Steffensen's method (for most functions, where calculation of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fixed Point (mathematics)
In mathematics, a fixed point (sometimes shortened to fixpoint), also known as an invariant point, is a value that does not change under a given transformation (mathematics), transformation. Specifically, for function (mathematics), functions, a fixed point is an element that is mapped to itself by the function. Any set of fixed points of a transformation is also an invariant set. Fixed point of a function Formally, is a fixed point of a function if belongs to both the domain of a function, domain and the codomain of , and . In particular, cannot have any fixed point if its domain is disjoint from its codomain. If is defined on the real numbers, it corresponds, in graphical terms, to a curve in the Euclidean plane, and each fixed-point corresponds to an intersection of the curve with the line , cf. picture. For example, if is defined on the real numbers by f(x) = x^2 - 3 x + 4, then 2 is a fixed point of , because . Not all functions have fixed points: for example, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
